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Stability of Nonlinear Systems Stability of Nonlinear Systems By Guanrong Chen Department of Electronic Engineering City University of Hong Kong Kowloon, Hong Kong SAR, China [email protected] In Encyclopedia of RF and Microwave Engineering, Wiley, New York, pp. 4881-4896, December 2004 S Continued STABILITY OF NONLINEAR SYSTEMS librium position of the system is stable [23]. The evolution of the fundamental concepts of system and trajectory sta- bilities then went through a long history, with many fruit- GUANRONG CHEN City University of Hong Kong ful advances and developments, until the celebrated Ph.D. Kowloon, Hong Kong, China thesis of A. M. Lyapunov, The General Problem of Motion Stability, finished in 1892 [21]. This monograph is so fun- damental that its ideas and techniques are virtually lead- 1. INTRODUCTION ing all kinds of basic research and applications regarding stabilities of dynamical systems today. In fact, not only A nonlinear system refers to a set of nonlinear equations dynamical behavior analysis in modern physics but also (algebraic, difference, differential, integral, functional, or controllers design in engineering systems depend on the abstract operator equations, or a combination of some of principles of Lyapunov’s stability theory. This article is these) used to describe a physical device or process that devoted to a brief description of the basic stability theory, otherwise cannot be clearly defined by a set of linear equa- criteria, and methodologies of Lyapunov, as well as a few tions of any kind. Dynamical system is used as a synonym related important stability concepts, for nonlinear dynam- for mathematical or physical system when the describing ical systems. equations represent evolution of a solution with time and, sometimes, with control inputs and/or other varying pa- 2. NONLINEAR SYSTEM PRELIMINARIES rameters as well. The theory of nonlinear dynamical systems, or nonlin- 2.1. Nonlinear Control Systems ear control systems if control inputs are involved, has been greatly advanced since the nineteenth century. Today, A continuous-time nonlinear control system is generally nonlinear control systems are used to describe a great va- described by a differential equation of the form riety of scientific and engineering phenomena ranging . from social, life, and physical sciences to engineering x ¼ fðx; t; uÞ; t 2½t0; 1Þ ð1Þ and technology. This theory has been applied to a broad spectrum of problems in physics, chemistry, mathematics, where x ¼ x(t) is the state of the system belonging to a n biology, medicine, economics, and various engineering dis- (usually bounded) region Ox R , u is the control input ciplines. vector belonging to another (usually bounded) region Ou Stability theory plays a central role in system engineer- Rm (often, m n), and f is a Lipschitz or continuously dif- ing, especially in the field of control systems and automa- ferentiable nonlinear function, so that the system has a tion, with regard to both dynamics and control. Stability of unique solution for each admissible control input and suit- a dynamical system, with or without control and distur- able initial condition x(t0) ¼ x0AOx. To indicate the time bance inputs, is a fundamental requirement for its prac- evolution and the dependence on the initial state x0, the tical value, particularly in most real-world applications. trajectory (or orbit) of a system state x(t) is sometimes Roughly speaking, stability means that the system out- denoted as jt(x0). puts and its internal signals are bounded within admissi- In control system (1), the initial time used is t0 0, ble limits (the so-called bounded-input/bounded-output unless otherwise indicated. The entire space Rn, to which stability) or, sometimes more strictly, the system outputs the system states belong, is called the state space. Associ- tend to an equilibrium state of interest (the so-called as- ated with the control system (1), there usually is an ob- ymptotic stability). Conceptually, there are different kinds servation or measurement equation of stabilities, among which three basic notions are the main concerns in nonlinear dynamics and control systems: y ¼ gðx; t; uÞð2Þ the stability of a system with respect to its equilibria, the orbital stability of a system output trajectory, and the where y ¼ yðtÞ2R‘ is the output of the system, 1‘n, structural stability of a system itself. and g is a continuous or smooth nonlinear function. When The basic concept of stability emerged from the study of both n, ‘>1, the system is called a multiinput/multioutput an equilibrium state of a mechanical system, dated back to (MIMO) system; while if n ¼ ‘ ¼ 1, it is called a single- as early as 1644, when E. Torricelli studied the equilibri- input/single-output (SISO) system. MISO and SIMO um of a rigid body under the natural force of gravity. The systems are similarly defined. classical stability theorem of G. Lagrange, formulated in In the discrete-time setting, a nonlinear control system 1788, is perhaps the best known result about stability of is described by a difference equation of the form conservative mechanical systems, which states that if the ( potential energy of a conservative system, currently at the xk þ 1 ¼ fðxk; k; ukÞ position of an isolated equilibrium and perhaps subject to k ¼ 0; 1; ÁÁÁ ð3Þ some simple constraints, has a minimum, then this equi- yk ¼ gðxk; k; ukÞ; 4881 4882 STABILITY OF NONLINEAR SYSTEMS where all notations are similarly defined. This article usu- at x~ ðtÞ. Then this Jacobian is also p-periodic: ally discusses only the control system (1), or the first equation of (3). In this case, the system state x is also Jðx~ ðt þ pÞÞ ¼ Jðx~ ðtÞÞ for all t 2½t0; 1Þ considered as the system output for simplicity. A special case of system (1), with or without control, is In this case, there always exist a p-periodic nonsingular said to be autonomous if the time variable t does not ap- matrix M(t) and a constant matrix Q such that the fun- pear separately (independently) from the state vector in damental solution matrix associated with the Jacobian the system function f. For example, with a state feedback Jðx~ ðtÞÞ is given by [4] control u(t) ¼ h(x(t)), this often is the situation. In this case, the system is usually written as FðtÞ¼MðtÞetQ . n x ¼ fðxÞ; xðt0Þ¼x0 2 R ð4Þ Here, the fundamental matrix F(t) consists of, as its col- umns, n linearly independent solution vectors of the linear . Otherwise, as (1) stands, the system is said to be nonau- equation x ¼ Jðx~ ðtÞÞx, with x(t0) ¼ x0. tonomous. The same terminology may be applied in the In the preceding, the eigenvalues of the constant ma- same way to discrete-time systems, although they may trix epQ are called the Floquet multipliers of the Jacobian. have different characteristics. The p-periodic solution x~ ðtÞ is called a hyperbolic periodic An equilibrium,orfixed point, of system (4), if it exists, orbit of the system if all its corresponding Floquet multi- à is a solution, x , of the algebraic equation pliers have nonzero real parts. Next, let D be a neighborhood of an equilibrium, xÃ,of à fðx Þ¼0 ð5Þ the autonomous system (4). A local stable and local unstable manifold of xà is defined by . It then follows from (4) and (5) that xà ¼ 0, which means that an equilibrium of a system must be a constant state. s à Wlocðx Þ¼fx 2DjjtðxÞ2D8tt0 and For the discrete-time case, an equilibrium of system ð9Þ Ã jtðxÞ!x as t !1g xk þ 1 ¼ fðxkÞ; k ¼ 0; 1; ... ð6Þ and is a solution, if it exists, of equation u à Wlocðx Þ¼fx 2DjjtðxÞ2D8tt0 and à ¼ ð ÃÞðÞ ð10Þ x f x 7 à jtðxÞ!x as t ! 1g An equilibrium is stable if some nearby trajectories of the system states, starting from various initial states, ap- respectively. Furthermore, a stable and unstable manifold à proach it; it is unstable if some nearby trajectories move of x is defined by away from it. The concept of system stability with respect s à s à O to an equilibrium will be precisely introduced in Section 3. W ðx Þ¼fx 2DjjtðxÞ\Wlocðx Þ fgð11Þ A control system is deterministic if there is a unique consequence to every change of the system parameters or and initial states. It is random or stochastic, if there is more u à u à O than one possible consequence for a change in its para- W ðx Þ¼fx 2DjjtðxÞ\Wlocðx Þ fgð12Þ meters or initial states according to some probability distribution [6]. This article deals only with deterministic respectively, where f denotes the empty set. For example, systems. the autonomous system ( . 2.2. Hyperbolic Equilibria and Their Manifolds x ¼ y . Consider the autonomous system (4). The Jacobian of this y ¼ xð1 À x2Þ system is defined by has a hyperbolic equilibrium (xÃ, yÃ) ¼ (0, 0). The local sta- @f ble and unstable manifolds of this equilibrium are illus- JðxÞ¼ ð8Þ @x trated by Fig. 1a, and the corresponding stable and unstable manifolds are visualized by Fig. 1b. Clearly, this is a matrix-valued function of time. If the Ja- A hyperbolic equilibrium only has stable and/or unsta- à cobian is evaluated at a constant state, say, x or x0, then it ble manifolds since its associated Jacobian has only stable à becomes a constant matrix determined by f and x or x0. and/or unstable eigenvalues. The dynamics of an autono- An equilibrium xà of system (4) is said to be hyperbolic mous system in a neighborhood of a hyperbolic equilibri- if all eigenvalues of the system Jacobian, evaluated at this um is quite simple—it has either stable (convergent) or equilibrium, have nonzero real parts.
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